Yes. In the next paragraph I will show that if $X$ is a Fréchet space (without requiring separability) then $X'_c$ with the compact-open topology is a $k$-space. As you note, this implies sequentiality if $X$ is separable.

The compact-open topology on $X'_c$ is the same as the finest topology agreeing with $\sigma(X'_c, X)$ on equicontinuous sets by the "Banach-Dieudonné" theorem (see Schaefer's *Topological Vector Spaces*, Chapter IV, Theorem 6.3). As Fréchet spaces are barrelled, all $\sigma(X'_c,X)$-bounded sets, and therefore all $\sigma(X'_c,X)$-compact sets, are equicontinuous. We also have, by the Bourbaki-Alaoglu theorem, that closed equicontinuous sets are $\sigma(X'_c,X)$-compact. Therefore the compact-open topology is the finest topology agreeing with $\sigma(X'_c,X)$ on compact sets, *i.e.* the Kelleyfication of the $\sigma(X'_c,X)$ topology, so $X'_c$ is a $k$-space.